Partition-crossing hypergraphs

Bujtás, Csilla and Tuza, Zsolt: Partition-crossing hypergraphs. Acta cybernetica, (23) 3. pp. 815-828. (2018)

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Abstract

For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . . , Xk) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Tur´an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.

Item Type: Article
Journal or Publication Title: Acta cybernetica
Date: 2018
Volume: 23
Number: 3
Page Range: pp. 815-828
ISSN: 0324-721X
Uncontrolled Keywords: Hipergráf, Gráfelmélet
Additional Information: Bibliogr.: p. 827-828. ; Összefoglalás angol nyelven
Date Deposited: 2018. Nov. 08. 08:21
Last Modified: 2018. Nov. 08. 08:21
URI: http://acta.bibl.u-szeged.hu/id/eprint/55679

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